On spectral distribution of kernel matrices related to radial basis functions

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• 作者：Andrew J. Wathen ; Shengxin Zhu
• 关键词：Eigenvalues ; Radial basis functions ; Spectral distribution ; Integral equation of the first kind ; 42A10 ; 45A25 ; 45B05 ; 45C05 ; 47A52 ; 47A75
• 刊名：Numerical Algorithms
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：70
• 期：4
• 页码：709-726
• 全文大小：416 KB
• 参考文献：1.Ball, K.: Eigenvalues of Euclidean distance matrices. J. Approx. Theory 68(1), 74-2 (1992). doi:10.-016/-021-9045(92)90101-S MathSciNet CrossRef MATH
  2.Ball, K., Sivakumar, N., Ward, J.D.: On the sensitivity of radial basis interpolation to minimal data separation distance. Constr. Approx. doi:8(4), 401-26 (1992). 10.-007/?BF01203461 MathSciNet CrossRef MATH
  3.Baxter, B.J.C.: Norm estimates for inverses of Toeplitz distance matrices. J. Approx. Theory 79(2), 222-42 (1994). doi:10.-006/?jath.-994.-126 MathSciNet CrossRef MATH
  4.Bernstein, S.: Sur la valeur les recherches récentes relatives à la meilleure approximation des fonctions continues par des polyn?mes. In: Proceedings 5th. Intern. Math. Congress, v. 1, vol. 1, pp 256-66 (1912)
  5.Buescu, J., Paix?o, A.: Eigenvalue distribution of positive definite kernels on unbounded domains. Integr. Equ. Oper. Theory 57(1), 19-1 (2007). doi:10.-007/?s00020-006-1445-1 CrossRef MATH
  6.Buescu, J., Paix?o, A.C.: Eigenvalue distribution of Mercer-like kernels. Math. Nachr. 280(9-10), 984-95 (2007). doi:10.-002/?mana.-00510530 MathSciNet CrossRef MATH
  7.Chang, C.H., Ha, C.W.: On eigenvalues of differentiable positive definite kernels. Integr. Equ. Oper. Theory 33(1), 1- (1999). doi:10.-007/?BF01203078 MathSciNet CrossRef MATH
  8.Cochran, J.A.: The analysis of linear integral equations. McGraw-Hill Book Co., New York (1972). McGraw-Hill Series in Modern Applied MathematicsMATH
  9.Fasshauer, G.E., McCourt, M.J.: Stable evaluation of Gaussian radial basis function interpolants. SIAM J. Sci. Comput. 34(2), A737–A762 (2012). doi:10.-137/-10824784 MathSciNet CrossRef MATH
  10.Ferreira, J.C., Menegatto, V.A.: Eigenvalues of integral operators defined by smooth positive definite kernels. Integr. Equ. Oper. Theory 64(1), 61-1 (2009). doi:10.-007/?s00020-009-1680-3 MathSciNet CrossRef MATH
  11.Fornberg, B., Larsson, E., Flyer, N.: Stable computations with Gaussian radial basis functions. SIAM J. Sci. Comput. 33(2), 869-92 (2011). doi:10.-137/-9076756X MathSciNet CrossRef MATH
  12.Fornberg, B., Piret, C.: A stable algorithm for flat radial basis functions on a sphere. SIAM J. Sci. Comput. 30(1), 60-0 (2007/08). doi:10.-137/-60671991
  13.Fornberg, B., Wright, G., Larsson, E.: Some observations regarding interpolants in the limit of flat radial basis functions. Comput. Math. Appl. 47(1), 37-5 (2004). doi:10.-016/?S0898-1221(04)90004-1 MathSciNet CrossRef MATH
  14.Fornberg, B., Zuev, J.: The Runge phenomenon and spatially variable shape parameters in RBF interpolation. Comput. Math. Appl. 54(3), 379-98 (2007). doi:10.-016/?j.?camwa.-007.-1.-28 MathSciNet CrossRef MATH
  15.Fredholm, I.: Sur une classe d’équations fonctionnelles. Acta Math. 27(1), 365-90 (1903). doi:10.-007/?BF02421317 MathSciNet CrossRef MATH
  16.Ha, C.W.: Eigenvalues of differentiable positive definite kernels. SIAM J. Math. Anal 17(2), 415-19 (1986). doi:10.-137/-517031 MathSciNet CrossRef MATH
  17.Hille, E., Tamarkin, J.D.: On the characteristic values of linear integral equations. Acta Math. 57(1), 1-6 (1931). doi:10.-007/?BF02403043 MathSciNet CrossRef
  18.de Hoog, F.R.: Review of Fredholm equations of the first kind, Application and numerical solution of integral equations (Proc. Sem., Australian Nat. Univ., Canberra, 1978), Monographs Textbooks Mech. Solids Fluids: Mech, Anal., vol. 6, pp. 119-34. Sijthoff & Noordhoff, Alphen aan den Rijn (1980)
  19.Horn, R.A., Johnson, C.R.: Matrix analysis. Cambridge University Press, Cambridge (1990). Corrected reprint of the 1985 originalMATH
  20.Katznelson, Y.: An introduction to harmonic analysis. John Wiley & Sons Inc., New York (1968)MATH
  21.Kress, R., 2nd edn.: Linear integral equations, Applied Mathematical Sciences, vol. 82. Springer, New York (1999)CrossRef
  22.Larsson, E., Fornberg, B.: Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions. Comput. Math. Appl. 49(1), 103-30 (2005). doi:10.-016/?j.?camwa.-005.-1.-10 MathSciNet CrossRef MATH
  23.Larsson, E., Lehto, E., Heryudono, E., Fornberg, B.: Stable computation of differentiation matrices and scattered node stencils based on Gaussian radial basis functions, Tech. Rep. 2012-020, Department of Information Technology, Uppsala University (2012)
  24.Little, G., Reade, J.B.: Eigenvalues of analytic kernels. SIAM J. Math. Anal. 15(1), 133-36 (1984). doi:10.-137/-515009 MathSciNet CrossRef MATH
  25.Micchelli, C.A.: Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constr. Approx. 2(1), 11-2 (1986). doi:10.-007/?BF01893414 MathSciNet CrossRef MATH
  26.Narcowich, F., Sivakumar, N., Ward, J.: On condition numbers associated with radial-function interpolation. J. Math. Anal. Appl. 186(2), 457-85 (1994). doi:10.-006/?jmaa.-994.-311 MathSciNet CrossRef MATH
  27.Narcowich, F., Ward, J.
• 作者单位：Andrew J. Wathen (1)
  Shengxin Zhu (2)
  
  1. Numerical Analysis Group, The University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, England
  2. Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, P.O.Box 8009, Beijing, 100088, China
  
• 刊物类别：Computer Science
• 刊物主题：Numeric Computing
  Algorithms
  Mathematics
  Algebra
  Theory of Computation
  
• 出版者：Springer U.S.
• ISSN：1572-9265

文摘

This paper focuses on spectral distribution of kernel matrices related to radial basis functions. By relating a contemporary finite-dimensional linear algebra problem to a classical problem on infinite-dimensional linear integral operator, the paper shows how the spectral distribution of a kernel matrix relates to the smoothness of the underlying kernel function. The asymptotic behaviour of the eigenvalues of a infinite-dimensional kernel operator are studied from a perspective of low rank approximation—approximating an integral operator in terms of Fourier series or Chebyshev series truncations. Further, we study how the spectral distribution of interpolation matrices of an infinite smooth kernel with flat limit depends on the geometric property of the underlying interpolation points. In particularly, the paper discusses the analytic eigenvalue distribution of Gaussian kernels, which has important application on stably computing of Gaussian radial basis functions. Keywords Eigenvalues Radial basis functions Spectral distribution Integral equation of the first kind